On the Biological Foundation of Risk Preferences Roberto Robattoyand Balázs Szentesz April 3, 2017
Abstract This paper considers a continuous-time biological model in which the growth rate of a population is determined by the risk attitude of its individuals. We consider choices over lotteries which determine the number of o¤spring and involve both idiosyncratic and aggregate risks. We distinguish between two types of aggregate risk: environmental variations and natural disasters. Environmental variations in‡uence the death and birth rates, while natural disasters result in instantaneous drops in population size. Our main result is a utility representation of the evolutionary optimal behavior. The utility is additively separable in the two types of aggregate risk. The term involving environmental variations is a von Neumann-Morgenstern utility which induces the same attitude towards both idiosyncratic and aggregate risk. The term involving disasters cannot be interpreted as an expected utility maximization and induces less tolerance towards aggregate risk.
1
Introduction
Most models in economics take preferences as given and derive the choices induced by those preferences. This paper does just the opposite. We entertain the hypothesis that choice behaviours are genetically determined and shaped by natural selection. The underlying individual preferences are then merely the representations of those evolutionary optimal choice behaviours. We work from the basic premise that in the long-run context of evolution, only the fastest-growing genes survive. As this paper focuses on risk preferences, we consider choices over lotteries that a¤ect the reproductive value of individuals. Our main result is a utility representation of the optimal choice behaviour. The corner stone of our analysis is a characterization of the long run dynamics of a population that inhabits a risky environment. Two types of aggregate risk are present, which we refer to as environmental variations and natural disasters. Environmental variations in‡uence the birth We have bene…ted from discussions with Je¤ Ely, Phil Reny, Arthur Robson and Larry Samuelson. of Finance, University of Wisconsin-Madison, Madison, WI, USA. E-mail:
[email protected]. z Department of Economics, London School of Economics, London, UK. E-mail:
[email protected]. y Department
1
and death rates of individuals, determining the rate of increase of the population size. Meanwhile, natural disasters cause discrete drops in population size, a¤ecting the level as opposed to the slope. The reproductive …tness of an individual is also subject to idiosyncratic risk, that is, conditional on an environmental variation or a natural disaster, the birth rate and the survival probability are still random variables. Our main result consists of a characterization of the asymptotic growth rate as a function of the ergodic distributions of the various types of risk. This function can then be interpreted as the utility representation of the evolutionary optimal choice behaviour over lotteries in an environment where the risk is determined by individuals’choices. In order to better explain our contribution and to contrast our results with those in existence, we …rst describe Robson’s (1996a) seminal paper on the evolution of risk preferences. Time is discrete and individuals live for one period. The number of o¤spring an individual has is determined by the realization of a lottery. A lottery, L, is described by a triple,
; G; fF ( j!)g!2
denotes the set of possible states of the world, G is the ergodic distribution on
, where
, and F ( j!) is
the distribution on the number of o¤spring if the state of the world is !. Conditional on !, the realization of F ( j!) is independent across individuals, so this element represents the idiosyncratic
component of the risk. The distribution G represents the aggregate risk, since ! determines the distribution of reproductive values in the population. Robson (1996a) shows that the asymptotic growth rate of the population is u (L) =
Z
log
Z
dF ( j!) dG (!) :
(1)
Since the gene inducing the choice of the lottery L grows at rate u (L) in the long run, Robson (1996a) interprets u as the utility representation of the evolutionary optimal behaviour. The main implication of (1) is that the evolved attitude towards risk depends strongly on whether the risk is idiosyncratic or aggregate. In fact, individuals will be relatively less tolerant of aggregate risk, as compared to idiosyncratic risk. In contrast to Robson (1996a), we consider a continuous time model, which allows us to distinguish between two types of aggregate risk: environmental variations and natural disasters. Recall that environmental shocks determine the rate of increase of the population size, while natural disasters cause instantaneous changes in the level. In discrete time, any aggregate shock necessarily results in a discrete change in the population size. It then becomes natural to ask whether the results obtained by Robson (1996a) apply only to natural disasters, or to aggregate risk in general. Our answer is that his results do not hold for environmental variations: the evolutionary optimal behaviour induces the same attitude towards both idiosyncratic risk and aggregate risk due to environmental variations. In order to state our representation theorem, let us describe our setup in detail. Time is continuous and population dynamics are determined by a lottery L = ( ; G; ; Fn ( j!) ; Fe ( j!)). The set
denotes the set of states of the world and G is the ergodic distribution on
function
:
. The
! R is the arrival rate of a disaster. The c.d.f. Fn ( j!) denotes the distribution of
2
the survival probability at the moment a natural disaster occurs. Fe ( j!) denotes the distribution
of the net birth-rate (birth-rate minus death-rate) of an individual conditional on !. Our main result is that the asymptotic growth rate, U (L), is Z Z Z Z U (L) = "dFe ("j!) dG (!) + (!) log
dFn ( j!) dG (!) :
(2)
The two additive terms in the utility representation (2) have very di¤erent economic interpretations. The …rst term, which is associated with environmental variations, is a standard von Neumann-Morgenstern representation with Bernoulli utility as the identity function. This implies that the choice induced by U depends only on the expected value of the net birth-rate; it doesn’t matter whether the source of the risk is aggregate or idiosyncratic. The second term, which is associated with natural disasters, is analogous to (1) of Robson (1996a). Indeed, if
(!)
1, which
corresponds to the frequency of discrete changes in Robson (1996a), the two expressions coincide. Let us again point out that this term is not a von Neumann-Morgenstern representation of risk attitude. Interestingly, this term is formally identical to a smooth ambiguity-averse preference representation where the ambiguity is determined by G and the ambiguity aversion is determined by the logarithm function.1 We think that the discrete model of Robson (1996a) is not only useful to describe natural disasters but also environments in which risk cannot naturally be characterized in terms of birthand death-rates, even in the absence of disasters. This may be exempli…ed by plant and animal species that reproduce periodically at particular times of the year. Consider an annual plant which reproduces only once in its lifetime and whose life cycle lasts one year. Although the plant might be exposed to various risks each day, this risk can intuitively be summarized using annual quantities, such as the probability that a seed survives until the reproductive season and the number of new seeds produced by the plant. As such, reproduction can be characterized in terms of factors rather than rates. The model of Robson (1996a) can be viewed as a description of the risk faced by such a plant in the case where the plant always survives until the reproductive season and produces seeds. Indeed, the appearance of the logarithmic function in the optimal choice criterion was …rst noticed in the context of annual organisms, see for example Cohen (1966). However, human reproduction occurs throughout the year rather than being con…ned to distinct breeding seasons.2 Approximating human population dynamics by a discrete model requires to group together all risks a¤ecting the population in a given time period and to characterize reproductive values in terms of factors. Our continuous-time model allows us to describe risks in more details and to express reproductive values in terms of birth- and death-rates. In the absence of natural disasters, this leads to an expected-utility representation (see the …rst term in (2)), in contrast to Robson’s non-expected utility representation in (1). We interpret our representation theorem as providing at least a partial evolutionary justi…cation for von Neumann-Morgenstern preferences. 1 For 2 Of
an axiomatic characterization of smooth ambiguity averse preferences, see Klibano¤ et al. (2005). course, risks faced by humans may be seasonal and this can be incorporated into the stochastic evolution of
the states in our continuous model.
3
Numerous papers use evolution to explain preferences. The …rst is probably Becker (1976), who adopts an evolutionary argument to explain altruism. Overviews of the theories on biological foundations of economic behaviour can be found in Robson (2001 and 2002). Robson (1996b) and Dekel and Scotchmer (1999) analyse optimal risk-taking strategic biological models, that is, an individual’s reproductive value is determined by not only her lottery but also by the lotteries of others. Our model does not involves strategic interactions; individuals only solve decision problems. Indeed, it is not clear to us how one might identify preferences and strategic components from equilibrium behaviour. The di¤erences between the e¤ects of idiosyncratic and aggregate risks on preferences are also emphasized by Robson and Samuelson (2009) in the context of time preferences. The authors consider an age-structured biological model and show that if the e¤ects of aggregate shocks on an individual’s survival probability do not depend on the age of the individual, then aggregate risk slows down population growth. In Section 8 of our Online Appendix, we extend our results to age-structured population. Among other things, we show that if the e¤ects of aggregate shocks on death rates are age-independent, the population growth is fully determined by the expected death rates and there is no distinction between the aggregate and idiosyncratic components.
2
Model
Time is continuous and is indexed by t 2 R+ . At each moment, a continuum of individuals make
up the population. The population dynamics are governed by the lottery L=
; G; f (!)g!2 ; fFe ( j!)g!2 ; fFn ( j!)g!2
:
(3)
R is the set of possible states of the world. Let ! t denote the state of the world at time t. The dynamic process f! t gt2R is a Markov process with unique ergodic distribution G.3 We assume that ! t is almost surely continuous in t almost everywhere.
Environmental variations.— The net birth-rate of an individual at time t is "t = bt bt denotes her birth rate and
t
t;
where
denotes her death rate. The variable "t 2 R is distributed according
to the conditional c.d.f. Fe ("t j! t ) and it is measurable in t almost surely. We allow the net birth
rate of an individual, "t , to change over time even if ! t stays constant. We assume that E ("j!) is a bounded function of !, and Fe ("j!) is uniformly continuous in !. The realizations of "0t s are assumed to be independent across individuals conditional on ! t . We assume that the Law of Large 3 In
particular, whenever
R
A
1dG (!) > 0 then for each ! 0 2 ; conditional on ! 0 = ! 0 : Z I (! t 2 A) dt = +1;
almost surely. That is, conditional on any initial condition, the process spends an in…nite amount of time in every positive G-measure set with probability one (see e.g. Glynn (1994) and Du¢ e and Glynn (2004)).
4
Numbers holds, and hence, at time t the population grows at rate Z r (! t ) "dFe ("j! t ) :
(4)
Natural disasters.— Natural disasters hit the population stochastically according to a nonhomogenous Poisson process. The arrival rate of the process at time t is :
1
(! t ) 2 [0; 1) where
! R+ is a bounded, measurable function in L ( ; G). If a natural disaster occurs at time
t, an individual survives with probability
t
2 [0; 1]4 , and Fn ( j! t ) denotes the distribution of
t
conditional on ! t . We assume that Fn ( j!) is uniformly continuous in !. We also assume that E ( j!) is uniformly bounded away from zero, that is, the population never goes extinct. Again, the realization of
t
is independent across individuals conditional on ! t . This assumption essentially
implies that the survival probability hence, the distribution of
t
t
is re-drawn at each moment a natural disaster occurs, and
is indeed independent of the previous history of disasters. We appeal
to the Law of Large Numbers once more and assume that the fraction of the population that survives a natural disaster is j (! t )
Z
dFn ( j! t ) :
We further assume that for each individual, "t and
t
(5)
are independent at each time t conditional
on ! t .
3
Results
In this section we analyze the speed of population growth. Let yt denote the size of the population at time t and let y0 be normalized to one. If there is a natural disaster at time t, yt denotes the size of the surviving population. The basic di¢ culty is that, due to aggregate shocks, the population does not grow at a steady state rate. Nevertheless, it is possible to characterize a growth rate, g, such that if t is large enough, the size of the population is approximately the same as if it were growing at a constant rate g, that is, egt :
yt
(6)
Such a growth rate g is called the continuously compounded growth rate of the population and is formally de…ned in Section 3.2. Next, we derive an expression for the law of motion of the population along a realized path of the random variables. In Section 3.2, we use this expression to prove our main result, which is a characterization of the continuously compounded growth rate in terms of the lottery (3). 4 For
our mathematical results to hold, we do not need
assumption for the sake of interpretation.
5
t
to be weakly less than one. We only make this
3.1
Law of motion of the population
Let N (t) denote the number of natural disasters which occurred between time zero and time t. In addition, let
i
2 [0; t], i 2 f1; :::; N (t)g ; denote the arrival time of the ith natural disaster.
Proposition 1 The size of the population at time t is given by: yt = exp
Z
N (t)
t
r (! s ) ds
0
where
QN (t) i=1
Y
j (! i ) ;
(7)
i=1
j (! i ) is de…ned to be one if N (t) = 0.
A notable property of the expression on the right-hand side of (7) is that it is multiplicatively separable in natural disasters and environmental variations. To illustrate this observation, let us consider a population that grows at a constant rate for a unit amount of time. Suppose that this population is hit by a disaster and half of the population dies. Then, irrespective of the exact time of the disaster, the population at the end of the time period will be half as large as it would have been if the disaster had not occurred. Of course, this argument presumes that the disaster leaves the net birth rate of the surviving population una¤ected. In our model, this assumption is satis…ed because "t and
t
are independent conditional on ! t .
Proof. We prove the proposition by induction with respect to the number of natural disasters, N (t). Suppose …rst that N (t) = 0. Then, by (4), the law of motion of the population is described by the following di¤erential equation between time zero and time t: :
y = y r (! ) .
(8)
The solution of this di¤erential equation is5 yt = exp
Z
t
r (! s ) ds ;
0
which is just the statement of the proposition for N (t) = 0. Suppose that the statement of the proposition is true for all t whenever N (t)
k and let us
assume that N (t) = k + 1. By the inductive hypothesis, lim !
y = exp
N (t)
<
At time
N (t)
Z
N (t) 1
Nt
r (! s ) ds
0
N (t)
j (! i ) .
i=1
there is a natural disaster and, by (5), only a fraction j !
survives. Hence, y
N (t)
= exp
Z
N (t)
of the population
N (t)
Nt
r (! s ) ds
0
5 The
Y
Y
j (! i ) :
i=1
solution exists because r is continuous and bounded, see Coddington and Levinson (1955), Chapter 2.
6
(9)
The law of motion of the population for
2[
N (t) ; t]
is again described by the di¤erential equation
(8) and initial condition (9). The solution is yt = y
Z
exp N (t)
!
t
r (! s ) ds N (t)
= exp
Z
N (t)
t
r (! s ) ds
0
Y
j (! i ) ;
i=1
where the second equality follows from (9).
3.2
Continuously compounded growth rate
Motivated by (6), we …rst provide a formal de…nition for the continuously compounded growth rate. De…nition 1 We call the number g 2 R the continuously compounded growth rate of the population
if
lim
t!1
log yt =g t
almost surely. In order to see that the continuously compounded growth rate is indeed a useful object in the evolutionary context, consider two populations, yt1 and yt2 , with corresponding compounded growth rates g1 and g2 . We show that if g1 > g2 then, asymptotically and with probability one, yt1 is going to be in…nitely large relative to yt2 . To see this, note that, by De…nition 1, log lim
t!1
t
yt1 yt2
log yt1 t!1 t
log yt2 = g1 t!1 t
= lim
lim
g2 > 0
almost surely. But this can only be the case if yt1 =yt2 converges to in…nity as t goes to in…nity with probability one. Next, we show that the continuously compounded growth rate exists and characterize it in terms of the lottery, L. This is our main result. Theorem 1 For almost all (f! t g ; fN (t)g)t2R ; Z Z Z log yt lim = "dFe ("j!) dG (!) + t!1 t
(!) log
Z
dFn ( j!) dG (!) :
Let us explain the basic idea of the proof. By Proposition 1, Rt PN (t) r (! s ) ds log yt 0 i=1 log j (! i ) lim = lim + lim ; t!1 t!1 t!1 t t t
(10)
(11)
where the second term in the right-hand side is de…ned to be zero if N (t) = 0. Since ! t is an ergodic process, both r (!) and log j (!) are also ergodic. As a consequence, the right-hand side of (10) is the sum of the time averages of two ergodic variables. Birkho¤’s Ergodic Theorem states that, under certain conditions, the time average of the realization of an ergodic variable converges
7
to the expected value of the variable with probability one, where the expectations are formed according to the ergodic distribution. In the proof, we argue that Birkho¤’s Ergodic Theorem is applicable, and show that the time average of r (!) and log j (!) converge to the …rst and second terms in the right-hand side of (10), respectively. Proof. First, consider the time average of r (! t ). Since the state of the world has an ergodic distribution and r is continuous, Birkho¤’s Ergodic Theorem implies that Rt Z r (! s ) ds 0 = r (!) dG (!) lim t!1 t almost surely6 . Substituting the de…nition of the function r in (4), we obtain Rt Z Z r (! s ) ds lim 0 = "dFe ("j!) dG (!) : t!1 t
(12)
We now turn to the second expression on the right-hand side of (11) and rewrite it as PN (t) N (t) i=1 log j (! i ) : t N (t) R If (!) dG (!) = 0 then = N (t) = 0 almost surely, so (12) implies (10). In what follows we R restrict attention to the case when (!) dG (!) > 0. We …rst show that Z N (t) lim = (!) dG (!) (13) t!1 t almost surely. To this end, assume that t 2 N for simplicity and de…ne Xi = N (i)
N (i
1) for
all i 2 N. Since N (t) is a non-homogenous Poisson process, the random variables X1 ; X2 ; ::: are Ri independent conditional on the realization of f! t gt2R+ , and EXi = V ar (Xi ) = i 1 (! ) d . By Kolmogorov’s Strong Law of Large Numbers7 lim
N (t)
t!1
Rt 0
(! ) d
t
= lim
t!1
Pt
i=1
h
Xi
Ri
i 1
t
(! ) d
i
=0
(14)
almost surely conditional on f! t gt2R+ . Finally, notice that Birkho¤’s Ergodic Theorem implies
that
1 t!1 t lim
Z
t
(! s ) ds =
0
Z
(!) dG (!)
(15)
almost surely. From (14) and (15) the equation in (13) follows. Next, we show that lim
t!1 6 See,
PN (t) i=1
log j (! i ) = N (t)
Z
log j (!) R
(!) dG (!) (! 0 ) dG (! 0 )
(16)
for instance, Doob (1953), Chapter XI, for a version of Birkho¤’s Ergodic Theorem for continuous-time
processes. 7 This theorem is indeed applicable since (See e.g. Feller (1968), Chapter X.7): Z i Z 1 X V ar [N (si ) N (si 1 )] 1 1 = lim 2 (! ) d = lim (!) dG (!) = 0 < 1: 2 i!1 i i!1 i i 0 i=1
8
almost surely. Recall that
denotes the time of the ith natural disaster. For all i 2 N, de…ne R Zi = log j (! i ). By (13) and (!) dG (!) > 0 it follows that there are in…nitely many disasters i
with probability one, so these variables are well-de…ned. The discrete time process fZi gi2N is also ^ is given by8 an ergodic Markov process and its ergodic distribution, G, R Z (!) dG (!) f!:log j(!)2Bg ^ R dG (!) = 0 (! ) dG (! 0 ) f!:log j(!)2Bg
for all Borel subset B of ( 1; 0]. Therefore, Birkho¤’s Ergodic Theorem implies that9 Pn Z i=1 Zi ^ (!) lim = log j (!) dG n!1 n
^ into the previous equation yields with probability one. Substituting the de…nitions of Zi and G (16). Finally, notice that from (13) and (16) it follows that
lim
t!1
PN (t) i=1
log j (! i ) = t
Z
(!) log j (!) dG (!) =
Z
(!) log
Z
dFn ( j! t ) dG (!)
(17)
almost surely, where the second equality is just (5). From equations (12) and (17) the statement of the theorem follows. As previously discussed, if the population corresponding to a particular gene has a higher continuously compounded growth rate than that of another population, it will eventually grow to be in…nitely larger than that other population. Therefore, if the level of environmental risk is determined by choices made by individuals, and that choice behaviour is genetic, the continuously compounded growth rate is the evolutionary optimal decision criterion. That is, only those genes which generate the largest continuously compounded growth rate survive in the long run. This leads us to interpret U (L) =
Z Z
"dFe ("j!) dG (!) +
Z
log
Z
dFn ( j!) (!) dG (!)
(18)
as the utility representation of the evolutionary optimal choice behaviour. One concern which might arise with respect to our analysis thus far pertains to our implicit assumption that the choice of a lottery is made once and for all, and determines the growth rate of the gene forever. We have not yet demonstrated that the same utility criterion is used to solve individual choice problems if the overall risk is determined by a combination of various decisions. In Section 6 of our Online Appendix, we take our analysis one step further and formalize the claim that the function U can indeed be interpreted as a utility function of the evolutionary optimal choice behaviour in this context. Interpretation.— The two terms on the right-hand side of (18) have very di¤erent behavioural implications. Observe that the lottery L can be viewed as the combination of two compound 8 See 9 See
Proposition 2 in Du¢ e and Glynn (2004). Corollary 1 of Proposition 2 of Du¢ e and Glynn (2004).
9
lotteries: (fFe ( j!)g! ; G) and (fFn ( j!)g! ; G), which are associated with environmental variations and natural disasters respectively. Equation (18) implies that a decision maker who maximizes U
reduces the compound lottery (fFe ( j!)g! ; G) to a simple one. That is, conditional on the risk
associated with natural disasters, choices will be based only on the expected rate of reproduction. Whether environmental variations are aggregate or idiosyncratic is irrelevant. Therefore, the …rst term is simply a standard von Neumann-Morgenstern representation. In sharp contrast, the second term does not correspond to expected utility maximization. In particular, the compound lottery (fFn ( j!)g! ; G) is not reduced to a simple one in the utility function U . This term is formally identical to the representation of smooth ambiguity-averse
preferences. In the context of ambiguity ( ; G) corresponds to the subjective state space and beliefs, Fn ( j!) describes uncertainty, and ambiguity-aversion is determined by the logarithmic
function. Since the logarithmic function is concave, this representation implies that the decision maker with utility function U is less tolerant towards aggregate risk than towards idiosyncratic risk. One notable feature of the function U is its additive separability in the risks due to environmental variations and natural disasters. This arises from the fact that yt is multiplicatively separable
in these two types of aggregate risk. As previously mentioned, this separability is due to our assumption that, for each individual, "t and
t
are independent conditional on ! t . We could relax
this independence assumption and still obtain an additive representation similar to (18). However, the c.d.f. Fe would be replaced by an ergodic distribution of the birth rates which would depend on, among other objects, Fn and . So, while the utility representation would still be additive, the term corresponding to environmental variations would depend on the risks faced due to disasters.
4
On the limit of the model of Robson (1996a)
Theorem 1 implies that environmental variations and natural disasters a¤ect growth in very different ways. We should perhaps shed some additional light on this observation, and we shall make it the objective of this section. To this end, we revisit Robson’s (1996a) discrete-time model and investigate the limit of a discrete time model as the length of the time intervals shrinks to zero. Indeed, it seems quite reasonable to suspect that environmental variations might be approximated arbitrarily well with a sequence of small natural disasters. In the context of a binary example, we show that, in the limit, the growth rate induced by a lottery does not depend on the decomposition of the risk. In this sense, there is no con‡ict between our results related to continuous-time models and the limiting behaviour of discrete-time models.10 1 0 In
the Online Appendix, we also consider the reverse of this exercise. That is, we establish a mapping from our
continuous-time models with only environmental variations to Robson’s (1996a) discrete-time models. We make use of this mapping to provide a clear explanation for the divergence between the continuous and discrete models with respect to di¤erent types of risk: in the continuous model, individuals make no distinction between aggregate and idiosyncratic risks, while in the corresponding discrete model they do.
10
Let us consider discrete populations facing two di¤erent lotteries; one which is purely idiosyncratic and the other purely aggregate. Robson (1996a) shows that the population facing the idiosyncratic lottery grows faster. In what follows, we show that as the length of a time interval approaches zero, the growth rates of the two populations converge.11 Suppose that time is discrete and indexed by t 2 N. Assume that
ergodic distribution of ! is given by Pr !
H
= Pr !
L
=
!H ; !L
and the
= 1=2. Each individual lives for one
period, and her number of o¤spring is the realization of a lottery. In each period t, the realization of the lottery is independent across individuals conditional on ! t . We consider the following two lotteries LA =
(
H L
if ! = ! H ; if ! = !
L
and LI =
(
H
with probability 1=2;
L
with probability 1=2,
where 1 < L < H. Note that the lottery LA involves only aggregate risk and the lottery LI involves only idiosyncratic risk. Also note that the risks induced by these two lotteries are the same from the viewpoint of a single individual; with probability one-half she produces H o¤spring, and with probability one-half she produces L. Let g (L) denote the compounded growth factor of the population corresponding to the lottery L (2 fLA ; LI g). The main result from Robson (1996a) implies
log g (LA ) =
log H + log L < log 2
H +L 2
= log g (LI ) ;
(19)
and hence, the population choosing LI grows faster than the other. In what follows, we shrink the length of the time intervals from one to consequences on the speed of population growth as
, and examine the
approaches zero. If the time intervals are
downscaled, and in each period individuals reproduce according to the lotteries LA and LI , the populations grow faster, and they explode as growth rates across di¤erent
goes to zero. Therefore, in order to compare
’s, the lotteries governing population dynamics must be rede…ned.
We shrink the intensity of the shocks speci…ed by the lotteries as
goes to zero. The idea of taking
the limit this way is to spread the e¤ect of the original lottery over many smaller time periods. We show that the per-unit-period growth rates of the populations corresponding the idiosyncratic and aggregate lotteries converge to the same value as
goes to zero. In other words, the distinction
between idiosyncratic and aggregate risk disappears. The intuition behind these observations can be explained as follows. If in the size of the population is also small within a
is small, the change
-long period. This means that the logarithmic
function in Robson’s (1996a) utility function, (1), can be approximated by a linear function arbitrarily well as
goes to zero. Therefore, the logarithmic function can be replaced by the linear
one in the limit. 1 1 We
are able to prove the same result in the general model of Robson (1996a) but we believe that the binary
example is su¢ cient to illustrate our point.
11
For each
, we de…ne the following two lotteries: ( ( H if ! = ! H ; H LA = and LI = L L if ! = ! L
with probability 1=2; with probability 1=2.
Observe that if the realization of the lottery is constant within a unit of time, the induced growth factor does not depend on
. To see why, note that if an individual produces H
-long period, she will have H
1=
o¤spring in each
= H genetic copies after one unit of time. We maintain the
assumption that the ergodic distribution of ! is given by Pr ! H = Pr ! L = 1=2.12 Let g denote the compounded growth factor of the population corresponding to L
2 LA ; LI
-long time period. Again, from Robson (1996a) it follows that log g Next, we take
LA =
log H
+ log L 2
< log
H
+L 2
= log g
LI .
L per
(20)
to zero and compare the growth factors per unit interval of the populations
corresponding to LA and LI . Note that if the growth factor per 1=
growth factor per unit interval is (g )
-long period is g
then the
. Consider …rst the population governed by the aggregate
lottery LA . By (20), lim log g
1=
LA = lim
!0
log H
!0
+ log L 2
=
log H + log L : 2
(21)
Comparing this expression with the left-hand side of (19), we conclude that the population facing the aggregate lottery is una¤ected by the time scaling. Consider now the population governed by the idiosyncratic lottery. Again by (20),
lim log g
1=
!0
log LI
= lim
H +L 2
!0
:
On the right-hand side, both the numerator and the denominator converge to zero. We apply L’Hopital’s rule to obtain lim log g !0
1=
LI
= lim
!1
1 H +L 2
H log H + L log L 2
=
where the second equality follows from the observation that both H goes to zero. From (21) and (22), we conclude that, as
1 (log H + log L) , 2 and L
(22)
converge to one as
goes to zero, the population facing
aggregate risk grows just as fast as the population facing only idiosyncratic risk. In the limit, do these discrete models become continuous-time models with only environmental variations? De…ne h and l such that H = eh and L = el and note that h and l are the rates corresponding to the factors H and L, respectively. The continuously compounded growth rates in 1 2 We
emphasize that this does not mean that the state of the world switches more and more frequently as
to zero. One can assume, for example, that ! switches with only probability
in each
the state of the world switches once per unit-period in expectation irrespective of
12
.
goes
-long time period. Then
(21) and (22) can then be written as (h + l) =2. In other words, the limits of LA and LI are lotteries involving only environmental variations according to which the net birth rate of an individual is h or l with equal probability. The limit of LA induces only aggregate risk and the limit of LI generates only idiosyncratic risk. By Theorem 1, this distiction, however, has no impact on the growth rate.
5
Conclusion
We believe endogenizing preferences to be an important research agenda, and that adopting the biological approach, as we have done, might prove to be a fruitful enterprise. It seems reasonable to hypothesize that evolution did not in‡uence only physical traits but also shaped choice behaviours. An advantage of this approach is that it has strong predictions about the relationship between fertility and choices which, in principle, can be tested empirically. We do recognize that many choice problems faced by individuals in modern times were unlikely to be faced in evolutionary times. Yet, we hypothesize that preferences, at least in part, are hardwired and that many choices made today re‡ect the evolutionary optimal behaviour.
References [1] Anscombe, F. J. and R. J. Aumann (1963): “A De…nition of Subjective Probability,” The Annals of Mathematical Statistics, 34, 199-205. [2] Becker, G. (1976): “Altruism, Egoism, and Genetic Fitness: Economics and Sociobiology,” Journal of Economic Literature, 14, 817-826. [3] Coddington, E.A. and N. Levinson (1955): Theory of Ordinary Di¤ erential Equations, McGraw-Hill. [4] Cohen, D (1966): “Optimizing Reproduction in a Randomly Varying Environment,” Journal of Theoretical Biology, 12, 119-129. [5] Dekel, E. and S. Scotchmer (1999): “On the Evolution of Attitudes towards Risk in WinnerTake-All Games,” Journal of Economic Theory, 87, 1999, 125–143. [6] Doob, J.L. (1953): Stochastic Processes, John Wiley and Sons. [7] Du¢ e, D. and P. Glynn (2004): “Estimation of Continuous-Time Markov Processes Sampled at Random Time Intervals,” Econometrica, 72, 1773-1808. [8] Feller, W. (1968): An Introduction to Probability Theory and Its Applications, Volume 1, John Wiley and Sons.
13
[9] Glynn, P. (1994): “Some Topics in Regenerative Steady-State Simulation,”Acta Applicandae Mathematicae, 34, 225-236. [10] Klibano¤, P., M. Marinacci and Sujoy Mukerji (2005): “A smooth model of decision making under ambiguity”, Econometrica, 73, 1849-1892. [11] Robson, A. (1996b): “The Evolution of Attitudes to Risk: Lottery Tickets and Relative Wealth”, Games and Economic Behavior, 14, 190-207. [12] Robson, A. (1996a): “A Biological Basis for Expected and Non-Expected Utility”, Journal of Economic Theory, 68, 397-424. [13] Robson, A. (2001): “The Biological Basis of Economic Behavior,” Journal of Economic Literature, 39, 11-33. [14] Robson, A. (2002): “Evolution and Human Nature,”Journal of Economic Perspectives, 16(2), 89-106. [15] Robson, A. and L. Samuelson (2009): “The Evolution of Time Preference with Aggregate Uncertainty,” The American Economic Review, 99(5), 1925-1953.
14
Online Appendix 6
Endogenous growth and utility representation
In this section, we introduce the stochastic arrival of choice problems. Each choice problem is a set of lotteries. Each of these lotteries is like L de…ned in (3). We de…ne a gene to be a choice behaviour, that is, a mapping from choice sets into choices. We then consider the dynamics of the population corresponding to a certain gene. At each moment in time, the dynamics of the population are determined by the lottery chosen in the most recent choice problem. We show that only those genes which behave as if they were maximizing the function U de…ned in (18) will survive. n
Let L denote an arbitrary set of lotteries of the form (3). The pair (flk gk=1 ; f
an environment where lk
L, jlk j < 1 and
A gene, c, in the environment
n (flk gk=1
;f
n k gk=1 )
is called
k (> 0) is the Poisson arrival rate of lk (k = 1; :::; n). n k gk=1 ) is a mapping from choice sets to elements of
these sets, that is, c : fl1 ; ::; ln g ! L, such that c (lk ) 2 lk for all k = 1; :::; n. Note that a gene is de…ned to be simply a choice behaviour on a restricted domain.
In what follows, we investigate the population growth corresponding to a certain gene, c. The n
population dynamics in the environment (flk gk=1 ; f
n k gk=1 )
are de…ned as follows. At t = 0, the
population is normalized to one and a choice problem is drawn, lk with probability
k=
n i=1 i .
If
t > 0 and there is an s < t, such that (i) no choice problem arrived during the time interval (s; t] and (ii) a choice problem l arrived at s, and c (l) = L then the dynamics of the population at time t are determined by the lottery L. We say that a gene survives evolution if its induced continuously compounded growth factor is at least as large as that of any other gene. n
Theorem 2 For each set of lotteries L and for each environment (flk gk=1 ; f ing gene, c , satis…es
c (lk ) 2 arg max U (L) , L2lk
n k gk=1 ) ;
the surviv(23)
for all k = 1; ::; n. Let us emphasize that the decision criterion in (23) is robust in the sense that the utility of a lottery is independent of the environment. That is, it depends neither on the other potential choice problems, nor on their frequencies of arrival. We have not allowed for idiosyncratic uncertainty in the arrival of choice problems. However, it would not be di¢ cult to introduce some such heterogeneity, and allow for the possibility that at some points in time di¤erent individuals of the same population face di¤erent problems. We might allow, for example, the population to consist of many colonies and the choice problems to arrive independently across colonies. In this case the size of the total population is simply the sum of the sizes of the colonies, so Theorem 2 remains valid.
15
n
n k gk=1 ) ;
Proof of Theorem 2. Fix an environment, (flk gk=1 ; f k
Lk =
; Gk ;
n
k
o (!)
!2
k
; Fek ( j!)
\
continuously compounded growth rate of the gene is n X
k 1
k=1
; Fnk ( j!)
!2
k
denote c (lk ). For notational convenience, assume that
and an arbitrary gene c. Let
m
!2
= f;g if k 6= m. We show that the
U (Lk ) :
+ ::: +
(24)
n
In other words, the asymptotic growth rate of the gene is the average of the continuously compounded growth rates induced by the chosen lotteries weighted by the frequencies of these choices. Note that the statement of the theorem follows from (24), because the fastest growing gene maximizes the summation in (24) pointwise, that is, it satis…es (23). We show that the dynamics of this population can be described by a lottery of the form (3). c
The set of possible states of the world,
, is [nk=1
k
. Since choice problems arrive independently
of ! and according to constant arrival rates, the lottery Lk determines the dynamics during a fraction of
k= ( 1
+ ::: +
n)
of the times. Therefore, the stationary distribution on
c
, Gc , is
given by k
Gc (!) = if ! 2
k
k
. Similarly, conditional on ! 2
1 + ::: +
Gk (!)
(25)
n
, the distribution of net birth rates, the arrival rate of
a natural disaster, and the distribution of survival probabilities are given by Fec ("j!) = Fek ("j!) ;
c
k
(!) =
(!) , and Fnc ( j!) = Fnk ( j!) ;
(26)
respectively. By (18), the continuously compounded growth rate of the population is given by Z Z Z Z c c "dFe ("j!) dG (!) + log dFnc ( j!) c (!) dGc (!) = =
Z n X
k=1 n X
k=1
!2
k
Z
"dFec ("j!) dGc (!) +
!2
k 1 + ::: +
Z
n
Z
!2
k
Z
log k
Z
"dFek ("j!) dGk (!) +
dFnc ( j!) Z
!2
log k
c
Z
(!) dGc (!) dFnk ( j!)
k
(!) dGk (!) ;
where the second equality follows from (25) and (26). Finally, notice that the last line of the previous equality chain is just (24).
7
Mapping continuous models into discrete models
There is a natural mapping from continuous-time models with only environmental variations into Robson’s (1996a) models. Indeed, one might simply evaluate the size of a continuously growing population at integer moments, and assume that all changes in population size happen at those
16
moments. In the continuous model, in order to make the evolutionary optimal choice, an individual only needs to know the expected net birth rates generated by the lotteries. In contrast, in the discrete model the growth rate of the population is not determined by the expected number of o¤spring; before making her choice, an individual needs to know whether the risk she faces is aggregate or idiosyncratic. It is straightforward to map a continuous model into a discrete model with similar aggregate population dynamics, but it is less clear how one might de…ne a discrete-time lottery which individuals will view as similar to continuous-time lotteries. One notable feature of our continuous model is the fact that, within a given time period, an individual might produce not only o¤spring but also grandchildren. Since children and grandchildren are identical, population growth is determined by the distribution of an individual’s descendants (as opposed to her o¤spring). Therefore, the distribution of o¤spring speci…ed by the discrete lottery should be equal to the one-period-ahead distribution of descendants in the continuous model. There is clearly no distinction between aggregate and idiosyncratic risks in the continuous model. The fact that such a distinction does exist in the discrete model might at …rst seem puzzling. One might argue, naively, that the expected number of descendants induced by a discrete lottery is pinned down by the expected net birth rate generated by the corresponding continuous lottery. It therefore does not seem possible that, in the continuous model, this expected value provides an individual with enough information to make a choice, while in the discrete model it does not. This reasoning is false. An individual’s net birth rate determines only the distribution of her number of o¤spring, and does not in fact determine her expected number of descendants. The distribution of her number of descendants also depends on whether the risk she faces is aggregate or idiosyncratic. In particular, if an individual’s birth rate is high, then her o¤spring’s birth rate is more likely to be high under aggregate risk than under idiosyncratic risk. Therefore, an individual with a high birth rate expects to have relatively more descendants if the risk she faces is aggregate. To resolve the puzzle, let us consider two continuous lotteries with the same expected net birth rate. On the one hand, the corresponding discrete time lotteries may have di¤erent expected values. In fact, a discrete lottery involving more aggregate risk will generate a higher expected number of descendants, which, in turn, promotes population growth. On the other hand, as Robson (1996a) showed, the larger the aggregate part of a discrete lottery, the slower the population will grow. It turns out that these two e¤ects cancel each other out, and the two discrete lotteries, with their di¤erent expected values, will generate the same growth rate. Indeed, each of these discrete-time lotteries yields the same utility a’la Robson (1996a) de…ned by (1). Let us now illustrate this argument with an example. Example. Suppose that
= ! H ; ! L and the state of the world can change only at integer
moments. Suppose that ! is redrawn at each t 2 N according to an i.i.d process de…ned by Pr ! H = Pr ! L = 1=2. Assume further that the net birth rate of an individual is constant
17
between integer moments and is determined by one of the following two lotteries: 8 8
Note that LcA involves only aggregate risk while LcI involves only idiosyncratic risk. In addition, the
expected net birth rate generated by both lotteries is log 2, and hence, each period, the populations corresponding to these lotteries double in expectation. The continuously compounded growth rate of both populations is log 2. In what follows, we characterize the discrete-time lotteries, LdA and LdI , corresponding to LcA and LcI . We show that the expected number of o¤spring induced by LdA is larger than that generated by LdI . However, we also show that the growth rates generated by these lotteries are the same, that is, u LdA = u LdI , where u is de…ned by (1). Consider …rst the case involving aggregate risk, that is, the lottery LcA . At each integer moment, the state of the world switches to ! L with probability one-half, and an individual’s net birth rate is set to zero. So, after one unit of time, her expected number of descendants (including herself) is one. With equal probability, ! switches to ! H , and the individual’s net birth rate becomes log 4. Because the risk is aggregate, within this time period the net birth rate of each of her descendants is also log 4. Therefore, her expected number of descendants is elog 4 = 4. To summarize, the discrete time lottery, LdA , corresponding to LcA is de…ned by ( 4 if ! = ! H , d LA = 1 if ! = ! L . Since the two possible states are equally likely, the expected number of descendants is 2:5. Let us next consider the case involving idiosyncratic risk, that is, the lottery LcI . At each integer moment, an individual’s net birth rate switches to zero with probability one-half, so after one unit of time her expected number of descendants is one (including herself). With the same probability, her net birth rate is set to log 4 at the beginning of the period. In addition, the net birth rate of each of her descendants born within that period is log 4 with probability one-half and zero otherwise. We do not characterize the distribution of the number of descendents but we observe that the expected number must equal three.13 Therefore, we conclude that the expected number of descendants generated by LdI is 2. Let us now examine the choice between LcA and LcI from a forward-looking perspective. Note that the lotteries LdA and LdI can be viewed as the summaries of the one-period-ahead consequences of LcA and LcI , respectively. On the one hand, an individual will have more descendants after one unit of time if she chooses the aggregate lottery LcA . Indeed, we have shown that ELdA = 2:5 > 1 3 The
reason is that an individual’s ex ante expected number of descendants is 2 because the population doubles
in each period and the risk is idiosyncratic. Since the expected number of descendant of an individual with birth rate zero is one, the expected number of descendants of an individual with birth rate log 2 must be 3.
18
2 = ELdI . On the other hand, aggregate risk slows down population growth (by the main result of Robson (1996a)). It turns out that these two e¤ects cancel each other out. Indeed, the two discrete lotteries corresponding to LcA and LcI induce the same growth rate. By the main result of Robson (1996a) (and our Theorem 2) the continuously compounded growth rates generated by LdI and LdA are both
1 1 log 4 + log 1 = log 2 = log ELdI = u LdI . 2 2 In other words, the long term consequences of a continuous lottery depend on the decomposition u LdA =
of the risk into idiosyncratic and aggregate parts. Nevertheless, an individual does not need to know these consequences. The expected net birth rate captures all the information she needs to make an optimal choice.
8
Age-structured Population
The discrete model of Robson (1996a) and our continuous model both involve population which are made up by identical adults. In particular, neither model involve individuals whose reproductive values vary with their ages. The objective of this section is to examine how the results described above generalize to arbitrary age-structured population. First, we generalize the discrete time model of Robson (1996a) to an arbitrary age-structured population. We derive the evolutionary optimal criterion and show that it does not have an expected utility representation. In particular, the logarithmic function appears in it just like in Robson’s (1996a) criterion. Then, we consider the age-structured version of our continuous-time model without natural disasters. We show that the evolutionary optimal behaviour maximizes the expected value of the average net birth-rate in the population and, in this sense, it has an expected utility representation. In this section, we abstract from natural disasters. Nevertheless, it is not hard to show that the conclusion of Robson (1996a) holds for natural disasters even if the population is age-structured, that is, the logarithmic function would appear in the optimal criterion.
8.1
The Generalization of Robson’s (1996a) Discrete Model
Time is discrete and is indexed by t 2 f0; 1; :::g. At each moment, a continuum of individuals make up the population. Each individual can live at most
dynamics are governed by the lottery L=
; G; fF ( j!; )g
!2 2f1;:::; g
2 N[ f1g periods. The population
; fH ( j!; )g
!2 2f1;:::; g
:
(27)
R is the set of possible states of the world. Let ! t denote the state of the world at time t. The dynamic process f! t gt2N is an ergodic Markov process with ergodic distribution G.
Reproduction.— The number of o¤spring of a -old individual at time t is "t 2 N+ . The 0-old
individuals are the newborns, so "t
0. For simplicity, we assume that newborns surely survive
19
until the age of one.14 We refer to individuals who are not newborns as adults. The variable "t is distributed according to the conditional c.d.f. F ( j! t ; ). We assume that E ("j!; ) is a bounded function of !, and F ( j!; ) is uniformly continuous in !. The realizations of "t ’s are assumed to
be independent across individuals conditional on ! t . We assume that the Law of Large Numbers holds, and hence, at time t the number of o¤spring produced by -old individuals is Z r (! t ) "dF ("j! t ; ) : Death.— A
-year old individual dies at time t with probability
t
survived until this age). Death occurs after reproduction. The variable
(conditional on having t
2 [0; 1] is distributed
according to the conditional c.d.f. H ( j! t ; ). Since individuals cannot live longer than
we assume that
t
1. The realizations of
t ’s
(28)
periods,
are assumed to be independent across individuals
conditional on ! t . We assume that the Law of Large Numbers holds, and hence, at time t the fraction of -year old individuals who dies is d (! t )
Z
dH ( j! t ; ) :
(29)
One might want to restrict attention to models where young individuals cannot produce o¤spring. Such a requirement is a restriction on the possible lotteries. For example, if one insists that individuals cannot reproduce prior to age , then only those lotteries should be considered 0 for all
which specify "t
.
The model of Robson (1996a) can be viewed as a special case of this model in the following two ways. First, suppose that individuals live for one period and then die, that is, 1 t
= 1 and
1. Second, suppose that individuals can live forever but adults are identical, that is,
and F ( j!; )
F ( j!), H ( j!; )
=1
H ( j!). Robson (1996a) interprets his model according to
(1) but the two models are mathematically identical.
The Evolutionary Optimal Criterion in the Discrete Model In general, the population growth in a given time period does not depend only on the state of the world, !, but also on the age-distribution of the population in that period. Furthermore, if there is aggregate risk, the age-distribution might change over time. We show that the age-distribution and the state of the world in a certain period fully determine the growth factor. Therefore, we rede…ne the state space and include the age-distribution as a new state variable. Observe that the age distribution and the realization of the state of the world in a certain period determine the age distribution in the next period by the lottery L. In other words, the lottery L de…nes a Markov process over the set of possible age-distributions. In what follows, we characterize the population growth in terms of the ergodic distribution over the new state space. To this end, let
denote the set of possible age-distributions of the adult population, that is, ( ) X = = f g 2f1;:::; g : =1 ; =1
1 4 Alternatively,
"t might denote the surviving o¤spring.
20
where
denotes the fraction of -old individuals. (The fraction of newborns are uniquely determined by the distribution of the adult population.) We de…ne e = as the new augmented e e state space. Let G denote the ergodic distribution over induced by the lottery L. The overall aggregate risk in an age structured population is captured by
e; G e
and not
by ( ; G). Indeed, given fF ( j!; )g and fH ( j!; )g, the correlation between the reproductive
values of two randomly selected adults is fully determined by (!; ) but it is not determined by !. The space ( ; G) determines only the correlation between the reproductive values of two e is in‡uenced randomly selected individuals of the same age. We emphasize that the distribution G
by the idiosyncratic component of the lottery, fF ( j!; )g and fH ( j!; )g, because the evolution
of the age distribution is a function of these objects. In the model of Robson (1996) without agestructure, the aggregate properties of population growth are determined by the state space, ( ; G), and the expected reproductive values, fE ("j!)g! . Similarly, in the age structured population, the
idiosyncratic components, F ( j!; ) and H ( j!; ), a¤ect the age distribution only through their expectations, E ("j!; ) and E ( j!; ).
Theorem 3 The continuously compounded growth rate of the population is # " Z Z Z X e (!; ) : "dF ("j!; ) + (1 ) dH ( j!; ) dG log
(30)
=1
Proof. Let yt and yt denote the size of the total adult population and the size of the
-old
population at time t. The number yt does not include the newborns at time t. We normalize y0 to be one. In addition, let
denote the fraction of -old individuals at time t, that is, yt =
t
Next, we establish the relationship between yt and yt t is the number newborns at t
yt
1r
(! t
1)
= yt
1
=1 1
= yt
1 t 1.
weakly older than 2 are those adults at time t yt =
=2
X
yt
1
X
1) ;
(1
d (! t
1 who survived that period, that is 1 ))
= yt
1
X
t 1
(1
d (! t
1 )) ,
=1
where the equality follows from yt X
(! t
The number of individuals at time t who are
=1
yt =
t 1r
=1
where the equality follows from yt X
The number of age-1 individuals at time
1; which is
X
yt1 =
1.
t yt .
1
= yt
yt = yt
=1
1
1 t 1.
X
t 1
To sum up the previous two displayed equalities: (r (! t
1)
+1
d (! t
=1
Using this formula recursively, yt =
tY 1
v=0
"
X
v
(r (! v ) + 1
=1
21
#
d (! v )) .
1 )) .
Hence, the continuously compounded growth rate of the population is Pt 1 P d (! v )) log yt v=0 log =1 v (r (! v ) + 1 lim = lim t!1 t!1 t t
Using Birkho¤’s Theorem, the right-hand side of the previous equality can be rewritten as " # Z X e (!; ) . log (r (!) + 1 d (!)) dG =1
Plugging (28) and (29) into the previous expression yields (30).
First, observe that this formula is a generalization of Robson’s (1996a) criterion, (1). We show this in both interpretations of Robson’s (1996a) model described above. Consider the …rst interpretation where individuals live for one period, that is, adults is degenerate:
1
1 t
1. Then the age distribution of
1. So, the expression (30) simpli…es to Z Z log "dF 1 ("j!) dG (!) ;
which is exactly the criterion of Robson (1996a). Consider now the second interpretation where = 1 and F ( j!; ) F ( j!), H ( j!; ) = H ( j!). Since does not a¤ect reproductive value, " # Z Z Z X e (!; ) = log "dF ("j!) + 1 dH ( j!) dG (31) = =
Z Z
log
"
log
=1
Z
X
=1
! Z
"dF ("j!) +
"dF ("j!) +
Z
1
Z
1
dH ( j!)
#
dH ( j!) dG (!) ;
e (!; ) dG
which is again the criterion of Robson (1996a). Note that the argument of the logarithmic function in (30) is the sum of the average number of o¤spring and the average survival probability of the population conditional on (!; ). Indeed, this quantity determines population growth conditional on (!; ). Observe that the continuously compounded growth rate is not just the expectation of this quantity. The criterion requires the computation of this quantity for each (!; ) and then integrates the logarithmic function of it according to the ergodic distribution. To summarize, the criterion (30) does not simply maximize the sum of the expected number of o¤spring and survival probabilities and, in this sense, it is inconsistent with expected utility maximization.
8.2
The Age-Structured Continuous Model
Time is continuous and is indexed by t 2 R+ . At each moment, a continuum of individuals make up the population. Each individual can live at most until the age of
dynamics are governed by the lottery L=
; G; fF ( j!; )g
!2 2[0; ]
22
; fH ( j!; )g
2 R+ [f1g. The population
!2 2[0; ]
:
(32)
R is the set of possible states of the world. Let ! t denote the state of the world at time t. The dynamic process f! t gt2R is an ergodic Markov process with ergodic distribution G. We assume that ! t is almost surely continuous in t almost everywhere.
Reproduction.— The birth-rate of a -old individual at time t is "t . The variable "t 2 R+ is
distributed according to the conditional c.d.f. F ("j! t ; ) and it is measurable in t almost surely and continuous in . We allow the birth rate of an individual to change over time even if ! t stays constant. We assume that E ("j!; ) is a bounded function of !, and F ("j!; ) is uniformly continuous in !. The realizations of "t 0 s are assumed to be independent across individuals conditional
on ! t . We assume that the Law of Large Numbers holds, and hence, at time t the -old individuals reproduce o¤spring at rate
Z
r (! t )
"dF ("j! t ; ) :
Death.— The death rate of a -old individual at time t is
(33) t.
The variable
t
2 R+ [ f1g is
distributed according to the conditional c.d.f. H ( j! t ; ) and it is measurable in t almost surely and periods, we assume that
continuous in . Since individuals cannot live longer than assume that E ( j!; ) is a bounded function of ! for in ! and continuous in . The realizations of
t ’s
t
1.15 We
< , and H ( j!; ) is uniformly continuous
are assumed to be independent across individuals
conditional on ! t . We assume that the Law of Large Numbers holds, and hence, at time t the -year old individuals die at rate d (! t )
Z
dH ("j! t ; ) :
(34)
We point out that our baseline model without natural disasters is a special case of this model where
= 1 and F ( j!; )
F ( j!) and H ( j!; )
H ( j!) for all . That is, individuals can
reach any age and their reproductive value is not a¤ected by their ages. Note that this corresponds to the second interpretation of the discrete model of Robson (1996a). The Evolutionary Optimal Criterion in the Continuous Model
Just like in the discrete model, the population growth in a given moment does not depend only on the state of the world, !, but also on the age-distribution of the population. Again, we rede…ne the state space and include the age-distribution. The lottery L de…nes a Markov process over the set of possible age-distributions. Again, we characterize the population growth in terms of the ergodic distribution over the new state space. To this end, let
denote the set of possible age-distributions, that is, ( ) Z = = f g 2[0; ] : 0; d =1 ; 0
denotes the fraction of -old individuals. We de…ne e = as the new augmented e denote the ergodic distribution over e induced by the lottery L. state space. Let G where
1 5 Alternatively,
we can just assume that individuals older than
23
cannot reproduce.
Theorem 4 The continuously compounded growth rate of the population is Z Z Z Z e (!; ) : "dF ("j!; ) dH ( j!; ) d dG
(35)
0
Proof. Let yt and yt denote the size of the total population and the size of the -old population at time t. We normalize y0 to be one. Using equations (33) and (34), the average net-birth rate in the population is Z d (! t )) d : t (r (! t ) =0
Therefore, similarly to (8), the population dynamics is governed by the following di¤erential equation: :
y t = yt The solution to this equation is yt = e
Z Rt
t
(r (! t )
d (! t )) d .
=0
z=0
R
=0
z (r
(! z ) d (! z ))d dz
:
Therefore, the continuously compounded growth rate can be written as Rt R (r (! z ) d d (! z )) d dz log yt = lim z=0 =0 z : lim t!1 t!1 t t The right-hand side is the time average of the net birth rate of the population. By Birkho¤’s Theorem, and plugging in (33) and (34), it is just (35). We show that this is a generalization of the part of our criterion which corresponds to environmental variations (the …rst term in (2)), where H ( j!) for all . Indeed, since Z Z = =
Z Z
Z Z
= 1 and F ("j!; )
F ("j!) and H ( j!; )
does not a¤ect reproductive values, (35) simpli…es to ! Z Z e (!; ) "dF ("j!) dH ( j!) d dG
"dF (" j!) "dF ("j!)
Z
Z
(36)
0
e (!; ) dH ( j!) dG
dH ( j!) dG (!) :
We next explain the extent to which the criterion, (35), is consistent with expected utility maximization. De…ne the myopic reproductive value of an individual at a given moment to be her net birth rate in that moment. These quantities are myopic because they do not take the individual’s and her descendants’future into account. We explain that, in the continuous model, the evolutionary optimal criterion, (35), maximizes the expected population average of the myopic reproductive value. To this end, note that for a given (!; ), the average reproductive value of the population is
Z
0
Z
"dF ("j!; )
24
Z
dH ( j!; ) d :
e Therefore, the expected myopic Furthermore, the state (!; ) is distributed according to G.
reproductive value is just (35). In other words, this criterion requires to maximize the expected myopic reproductive value of the population and it does not matter whether this expectation is
based on aggregate risk (!; ) or idiosyncratic risk. In sharp contrast, as we explained above, the optimal criterion in the discrete model, (30), is di¤erent from the expected myopic reproductive value and it matters whether the myopic reproductive value is in‡uenced by aggregate risk or idiosyncratic risk. Finally, we explain that (35) can be interpreted as the expected myopic reproductive value of e (!; ) is a a randomly picked individual from the population at a random moment. Note that G
distribution over age distributions for each !. Since a distribution over distribution is a distribution, e (!; ) generates a distribution over [0; ] for each !. Let G (!; ) denote this distribution, that G is, G (!; ) is the distribution of the age of a randomly picked individual conditional on !. Then (35) can be rewritten as Z Z
Z
"dF ("j!; )
Z
dH ( j!; ) dG (!; ) ,
which is indeed the myopic reproductive value of a randomly picked individual.
8.3
Intertemporal E¤ects
In the context of age-structured population, lotteries specify risk over streams of reproductive values. As a consequence, the preferences over lotteries de…ned by (27) in the discrete model and by (32) in the continuous model do not only re‡ect the attitude towards risk but also the attitude towards intertemporal trade-o¤s. To elabotrate on the lotteries’ intertemporal e¤ects, note that the realization of the state of the world at time t, ! t , does not only in‡uence the growth rate of the population at time t but also the growth rates in future periods. The reason is that, in general, ! t a¤ects the age distribution of the population in future periods. In turn, as we have shown, the age-distribution a¤ects the growth rate. Therefore, by including the potential age distributions into the state space we managed to restrict our attention to risk and abstract away from intertemporal trade-o¤s. Indeed, the growth rate of the population at time t is fully determined by the realization (! t ;
t ).
Previous realizations of the augmented state a¤ect the
growth rate only at time t indirectly, through their stochastic e¤ects on (! t ; t ). This risk is e We explained above that (35) can be interpreted as the captured by the ergodic distribution G.
expected net birth-rate of a randomly selected individual at a random moment in time. In other
words, augmenting the state space allows one to collapse the intertemporal trade-o¤s to a risk about age.
Another, and perhaps more natural, way to eliminate time preferences from our analysis is to restrict attention to choices over a set of lotteries which do not involve intertemportal tradeo¤s. An example for such a set include lotteries which only di¤er in age-independent survival probabilities (in the discrete model) or death-rates (in the continuous model). The reason is
25
that the risk over age-independent survival probabilities and death-rates has no impact on the age distribution in the population and hence, the choices re‡ect to risk preferences over the ageindependent risk. As we mentioned in the Introduction, Robson and Samuelson (2009) examine choices over this set of lotteries in a discrete model. They show that adding aggregate risk without changing expected individual survival probabilities slows down population growth. In other words, optimal choices do not depend only on expected age-independent survival probabilities but also on the decomposition of risk into aggregate and idiosyncratic components. In contrast, we next show that, in our continuous model, the population growth is fully determined by the expected death-rates and there is no distinction between the aggregate and idiosyncratic components. In other words, adding aggregate risk to death-rates has no impact as long as it does not change expected death-rates. Proposition 2 Consider two lotteries Li = such that
R
; G; fF ( j!; )g
dH1 ( j!; ) =
R
!2 2[0; ]
; fHi ( j!; )g
!2 2[0; ]
dH2 ( j!; ) + d (!) for some d :
the two lotteries induce the same growth rate.
; i = f1; 2g ;
! R. If
R
d (!) dG (!) = 0, then
Proof. Since age-independent death rates have no e¤ect on the age distribution, both lotteries generate the same Markov process over age distributions. This implies that the ergordic distributions generated by L1 and L2 over the augmented state space are the same. Therefore, by (35), L1 R induces a larger continuous compounded growth rate than L2 does if and only if d (!) dG (!) 0. R In particular, if d (!) dG (!) = 0, then the two lotteries induce the same growth rate. In the next section, we take this analysis one step further. We consider an arbitrary set of
lotteries which generate the same Markov process over age distributions. We will show that the optimal choice behaviour over this set satis…es the independence axiom in the continuous model.
8.4
The Independence of Axiom
Next we discuss the choice behavioural di¤erence between the discrete and continuous models. We explain that the independence axiom is violated according to the discrete criterion but it is satis…ed according to the continuous criterion. We revisit the framework of subjective utility theory of Anscombe and Aumann (1963). We associate the aggregate state space of our biological model to the subjective state space of Anscombe and Aumann (1963), and we associate the idiosyncratic risk to known uncertainty in Anscombe and Aumann (1963). First, we explain the choice behavioural di¤erence between the original models of Robson (1996a) and our baseline model without agestructure. We state the independence axiom for lotteries with the same state space ( ; G) and argue that it is satis…ed by the criterion (36) and violated by Robson’s (1996a) criterion, (1). Then, we turn our attention to age-structured population. As we explained above, in general, lotteries
26
have intertemporal e¤ects. In order to isolate risk preferences from time preferences, we state the e . We then independence axiom for lotteries which generate the same aggregate state space e ; G
show that the continuous criterion, (35), satis…es the independence axiom whereas the discrete one, (30), violates it. Finally, we explain that the violation of the independence axiom when it is stated e , can be solely due to intertemporal for lotteries with di¤erent augmented state spaces, e ; G
trade-o¤s and might not re‡ect to risk preferences. Indeed, we show that the independence axiom can be violated by lotteries even if they do not involve any risk. Without the age structure.— Consider the lottery L =
; G; fF ( j!)g!2 ; fH ( j!)g!2
model where adult individuals are identical. The myopic reproductive values, "
in a
, correspond to
the prizes in Anscombe and Aumann (1963). The aggregate state space ( ; G) corresponds to the subjective state space and beliefs in Anscombe and Aumann (1963). Finally, the idiosyncratic risk, fF ( j!)g!2 ; fH ( j!)g!2 , correspond to objective randomization conditional on the realization
of the state. Just like Anscombe and Aumann (1963), we de…ne the convex combination of two lotteries, L1 =
; G; fF1 ( j!)g!2 ; fH1 ( j!)g!2
; L2 =
; G; fF2 ( j!)g!2 ; fH2 ( j!)g!2
;
as follows: L1
(1
) L2 =
; G; f F1 ( j!) + (1
) F2 ( j!)g!2 ; f H1 ( j!) + (1
) H2 ( j!)g!2
.
That is, for each !, an individual’s myopic reproductive value is determined by F1 ( j!) and H1 ( j!) with probability
and it is determined by F2 ( j!) and H2 ( j!) with probability (1
).16
The Independence Axiom: Let L1 ; L2 and L3 be lotteries corresponding to the same state space ( ; G). Then for all 2 [0; 1] ;
L1
L2 , L1
(1
) L3
Note that our continuous criterion, (36), is linear in
L2
(1
) L3 :
(37)
and, hence, this criterion satis…es this
axiom. In contrast, the discrete criterion of Robson (1996), (31), violates this criterion. Indeed, the discrete criterion corresponds to a smooth ambiguity averse preference where the ambiguity aversion is determined by the logarithmic function. Age Structure.— Just like in the models without age structure, we state the independence axiom for lotteries which correspond to the same aggregate risk. As we explained above, the e , so we require the independence axiom to hold overall aggregate risk is determined by e ; G
for lotteries corresponding to the same augmented state space. To this end, …x ( ; G) and let D denote the set of Markov processes on 1 6 In
Anscombe and Aumann (1963), the parameter
. For each d 2 D, let L (d) denote the set of
represents an objective uncertainty, hence, we assume that
it corresponds to idiosyncratic risk in our biological model.
27
those lotteries which specify ( ; G) and generate generate the Markov process d over
. Note
that if L1 and L2 correspond to the same state space ( ; G) and EL1 ("j!; ) = EL2 ("j!; ) and EL1 ( j!; ) = EL2 ( j!; ) then they de…ne the same Markov process over
, that is,
L1 ; L2 2 L (d) for some d. Two lotteries also de…ne the same dynamic process over the augmented state space if they di¤er only in a state-dependent but age-independent death rates. Now, the independence axiom for age structured population can be stated as follows: For each d 2 D, and for all L1 ; L2 ; L3 2 L (d), (37) holds.
Theorem 5 The continuous criterion (35) satis…es the Independence Axiom (37). Proof. Fix lotteries L1 ; L2 and L3 such that they all specify ( ; G) and generate the Markov process d over
. Suppose that L1 Z Z Z Z
Z
0
Z
0
L2 , that is, Z
"dF1 ("j!; )
Z
"dF2 ("j!; )
This inequality is equivalent to Z Z
0
+ (1 Z Z + (1
Z
)
Z Z
)
Z Z
Z
0
Z
0
"dF1 ("j!; )
0
e (!; ) . dH2 ( j!; ) d dG
e (!; ) dH1 ( j!; ) d dG Z
"dF3 ("j!; )
"dF2 ("j!; ) Z
Z
e (!; ) dH1 ( j!; ) d dG
Z
e (!; ) dH3 ( j!; ) d dG
e (!; ) dH2 ( j!; ) d dG Z
"dF3 ("j!; )
e (!; ) . dH3 ( j!; ) d dG
Since L1 ; L2 ; L3 2 L (d) ; the previous inequality is equivalent to Z Z
Z Z
0
0
Z Z
"d [ F1 + (1
) F3 ] ("j!; )
"d [ F2 + (1
) F3 ] ("j!; )
This last inequality is just L1
(1
) L3
Z Z
L2
d [ H1 + (1 d [ H2 + (1 (1
e (!; ) )] H3 ( j!; ) d dG
e (!; ) : )] H3 ( j!; ) d dG
) L3 , hence, (37) indeed holds.
In contrast, the discrete criterion, (30), violates the independence axiom. In order to see this, recall that we have shown that the discrete criterion (30) is the generalization of Robson’s (1996a) criterion, (1). Furthermore, we have already argued that this latter criterion violates (37). Finally, we explain that, in an age structured population, the violation of the independence e axiom for lotteries with the same state space ( ; G) but di¤erent augmented state space e ; G might have nothing to do with risk preferences. In the following example, we consider lotteries
28
which do not involve any risk. We show that, due to the intertemporal trade-o¤ described in the previous section, the independence axiom is violated. Example. Consider the following two discrete time lotteries: Li =
; G; fFi ( j!; )g
; fHi ( j!; )g
2f1;2g
2f1;2g
i 2 f1; 2g :
Neither of the lotteries involve any risk and, according to each lottery, individuals die at age 2, that is,
= 2.
L1 : individuals surely survive until the age of two. They reproduce one o¤spring at each age. Formally, for each ! 2 if "
1 for
= 1; 2.
, H1 ( j!; 1) = 1 for all
0, F1 ("j!; ) = 0 if " < 1 and F1 ("j!; ) = 1
L2 : individuals reproduce one o¤spring at age 1 and then they die. Had they survived until the age of 2, they reproduce four o¤spring. Formally, for each ! 2 F2 ("j!; 1) = 0 if " < 1 and F2 ("j!; 1) = 1 if " if "
, H2 ( j!; 1) = 0 for all
1,
1, and F2 ("j!; 2) = 0 if " < 4 and F2 ("j!; 2) = 1
4. The advantage of the population governed by L1 is that individuals reproduce at age two.
Therefore, the population evolving according to L1 grows faster than the population evolving according to L2 , that is, L1
L2 . Next, we argue that L1
1 L1 2
1 L1 2
1 L2 2
1 L1 ; 2
(38)
that is, the independence axiom is violated. To see this, note that according to
1 2 L2
1 2 L1 ,
each individual survives until the age of two with probability one half and reproduces four or one o¤spring with equal probabilities. Therefore, from the perspective of a newborn, each individual reproduces 1:25 o¤spring at age 2 in expectation. According to L1 , each individual has only one o¤spring at age 2. Since this is the only di¤erence between L1 and 21 L2
1 2 L1 ,
we conclude that
(38) holds. Note that neither of the individual lotteries above involve any risk. Why then is the independence axiom violated? Combining L1 with L2 has the following intertemporal e¤ect on an individual. If an individual at age 1 is lucky, her survival probability is determined by L1 , and then she survives until the age of 2. (She would die if her survival probability is determined by L2 .) If at age two she is lucky again, the number of her o¤spring is determined by L2 and reproduces four o¤spring (as opposed to just one according to L1 ). In other words, an individual might have high reproductive value at age two because her survival probability was determined by L1 in the previous period. To put it di¤erently, note that the reproductive value of exactly half of the population is determined by L2 in each period. However, the age distribution of this part of the population is di¤erent from any age distribution which can be generated by L2 alone. In this stark example, the age distribution generated by L2 is degenerate,
1
1 and
2
0. The
reproductive value of an individual at age 2 is irrelevant because individuals never survive until the
29
age of two. On the other hand, when combining L2 with L1 , a signi…cant fraction of the population survives until age two and reproduces according to L2 . In other words, there is no sense in which L1 should be considered an irrelevant alternative when combining with L2 . The alternative L1 is very relevant because it has an impact on the age distribution on which L2 operates. In general, the combination of two lotteries can result in age distributions which never arise according to the individual lotteries. The population growth is determined by evaluating the lotteries on these new age distributions. To summarize, the violation of the independence axiom of arbitrary lotteries can be due to the intertemporal e¤ects of the lotteries. In contrast, the combination of two lotteries, L1 and L2 , such that L1 ; L2 2 L (d), does not require the consideration of intertemporal e¤ects. Indeed, consider the combination of these two lotteries, In each period,
L1
(1
) L2 .
fraction of the population evolves according to L1 . In addition, the age distri-
bution of the population is exactly the same as if the population would have evolved according to L1 . Therefore, the growth rate of this population at time t is the same as if the population would have evolved according to L1 before t and the reproductive values would be determined by L1
(1
) L2 at time t. In this sense, L2 has no intertemporal e¤ect on L1 .
30
